Disk Surface Density Profiles Research Articles

A Recipe for Eccentricity and Inclination Damping for Partial-gap Opening Planets in 3D Disks

In a previous paper, we showed that, like the migration speed, the eccentricity damping efficiency is modulated linearly by the depth of the partial gap a planet carves in the disk surface density profile, resulting in less efficient e-damping compared to the prescription commonly used in population synthesis works. Here, we extend our analysis to 3D, refining our e-damping formula and studying how the inclination damping efficiency is also affected. We perform high-resolution 3D locally isothermal hydrodynamical simulations of planets with varying masses embedded in disks with varying aspect ratios and viscosities. We extract the gap profile and orbital damping timescales for fixed eccentricities and inclinations up to the disk scale height. The limit in gap depths below which vortices appear, in the low-viscosity case, happens roughly at the transition between classical type-I and type-II migration regimes. The orbital damping timescales can be described by two linear trends with a break around gap depths ∼80% and with slopes and intercepts depending on the eccentricity and inclination. These trends are understood on physical grounds and are reproduced by simple fitting formulas whose error is within the typical uncertainty of type-I torque formulas. Thus, our recipes for the gap depth and orbital damping efficiencies yield a simple description for planet–disk interactions to use in N-body codes in the case of partial-gap opening planets that is consistent with high-resolution 3D hydrosimulations. Finally, we show examples of how our novel orbital damping prescription can affect the outcome of population synthesis experiments.

DPNNet-2.0. I. Finding Hidden Planets from Simulated Images of Protoplanetary Disk Gaps

Abstract The observed substructures, like annular gaps, in dust emissions from protoplanetary disks are often interpreted as signatures of embedded planets. Fitting a model of planetary gaps to these observed features using customized simulations or empirical relations can reveal the characteristics of the hidden planets. However, customized fitting is often impractical owing to the increasing sample size and the complexity of disk–planet interaction. In this paper we introduce the architecture of DPNNet-2.0, second in the series after DPNNet, designed using a convolutional neural network (CNN, specifically ResNet50 here) for predicting exoplanet masses directly from simulated images of protoplanetary disks hosting a single planet. DPNNet-2.0 additionally consists of a multi-input framework that uses both a CNN and multilayer perceptron (a class of artificial neural network) for processing image and disk parameters simultaneously. This enables DPNNet-2.0 to be trained using images directly, with the added option of considering disk parameters (disk viscosities, disk temperatures, disk surface-density profiles, dust abundances, and particle Stokes numbers) generated from disk–planet hydrodynamic simulations as inputs. This work provides the required framework and is the first step toward the use of computer vision (implementing CNNs) to directly extract the mass of an exoplanet from planetary gaps observed in dust surface-density maps by telescopes such as the Atacama Large Millimeter/submillimeter Array.

Constraining the Chemical Signatures and the Outburst Mechanism of the Class 0 Protostar HOPS 383

Abstract We present observations toward HOPS 383, the first known outbursting Class 0 protostar located within the Orion molecular cloud using the Atacama Large Millimeter/submillimeter Array (ALMA), Very Large Array (VLA), and Submillimeter Array (SMA). The SMA observations reveal envelope scale continuum and molecular line emission surrounding HOPS 383 at 0.85, 1.1, and 1.3 mm. The images show that HCO+ and H13CO+ peaks on or near the continuum, while N2H+ is reduced at the same position. This reflects the underlying chemistry where CO evaporating close to the protostar destroys N2H+ while forming HCO+. We also observe the molecular outflow traced by 12CO ( ) and ( ). A disk is resolved in the ALMA 0.87 mm dust continuum, orthogonal to the outflow direction, with an apparent radius of ∼62 au. Radiative transfer modeling of the continuum gives disk masses of 0.02 M ⊙ when fit to the ALMA visibilities. The models including VLA 8 mm data indicate that the disk mass could be up to a factor of 10 larger due to lower dust opacity at longer wavelengths. The disk temperature and surface density profiles from the modeling, and an assumed protostar mass of 0.5 M ⊙ suggest that the Toomre Q parameter < 1 before the outburst, making gravitational instability a viable mechanism to explain outbursts at an early age if the disk is sufficiently massive.

A Machine Learning Model to Infer Planet Masses from Gaps Observed in Protoplanetary Disks

Abstract Observations of bright protoplanetary disks often show annular gaps in their dust emission. One interpretation of these gaps is disk–planet interaction. If so, fitting models of planetary gaps to observed protoplanetary disk gaps can reveal the presence of hidden planets. However, future surveys are expected to produce an ever-increasing number of protoplanetary disks with gaps. In this case, performing a customized fitting for each target becomes impractical owing to the complexity of disk–planet interaction. To this end, we introduce Disk Planet Neural Network (DPNNet), an efficient model of planetary gaps by exploiting the power of machine learning. We train a deep neural network with a large number of dusty disk–planet hydrodynamic simulations across a range of planet masses, disk temperatures, disk viscosities, disk surface density profiles, particle Stokes numbers, and dust abundances. The network can then be deployed to extract the planet mass for a given gap morphology. In this work, first in a series, we focus on the basic concepts of our machine learning framework. We demonstrate its utility by applying it to the dust gaps observed in the protoplanetary disk around HL Tau at 10, 30, and 80 au. Our network predicts planet masses of 80 M ⊕, 63 M ⊕, and 70 M ⊕, respectively, which are comparable to those from other studies based on specialized simulations. We discuss the key advantages of our DPNNet in its flexibility to incorporate new physics as well as any number of parameters and predictions, in addition to its potential to ultimately replace hydrodynamical simulations for disk observers and modelers.

New Constraints on the Dust and Gas Distribution in the LkCa 15 Disk from ALMA

Abstract We search a large parameter space of the LkCa 15's disk density profile to fit its observed radial intensity profile of 12CO (J = 3–2) obtained from Atacama Large Millimeter/submillimeter Array. The best-fit model within the parameter space has a disk mass of 0.1 M ⊙ (using an abundance ratio of 12CO/H2 = 1.4 × 10−4 in mass), an inner cavity of 45 au in radius, an outer edge at ∼600 au, and a disk surface density profile that follows a power law of the form ρ r ∝ r −4. For the disk density profiles that can lead to a small reduced χ 2 of the goodness-of-fit, we find that there is a clear linear correlation between the disk mass and the power-law index, γ, in the equation of the disk density profile. This suggests that the 12CO disk of LkCa 15 is optically thick, and we can fit its 12CO radial intensity profile using either a lower disk mass with a smaller γ or a higher disk mass with a bigger γ. By comparing the 12CO channel maps of the best-fit model with disk models with higher or lower masses, we find that a disk mass of ∼0.1 M ⊙ can best reproduce the observed morphology of the 12CO channel maps. The dust continuum map at 0.87 mm of the LkCa 15 disk shows an inner cavity of the similar size of the best-fit gas model, but its outer edge is at ∼200 au, which is much smaller than the fitted gas disk. Such a discrepancy between the outer edges of the gas and dust disks is consistent with dust drifting and trapping models.

Probing the protoplanetary disk gas surface density distribution with 13CO emission

Context. How protoplanetary disks evolve is still an unsolved problem where different processes may be involved. Depending on the process, the disk gas surface density distribution Σgas may be very different and this could have diverse implications for planet formation. Together with the total disk mass, it is key to constrain Σgas as function of disk radius R from observational measurements. Aims. In this work we investigate whether spatially resolved observations of rarer CO isotopologues, such as 13CO, may be good tracers of the gas surface density distribution in disks. Methods. Physical-chemical disk models with different input Σgas(R) were run, taking into account CO freeze-out and isotope-selective photodissociation. The input disk surface density profiles were compared with the simulated 13CO intensity radial profiles to check whether and where the two follow each other. Results. For each combination of disk parameters, there is always an intermediate region in the disk where the slope of the 13CO radial emission profile and Σgas(R) coincide. In the inner part of the disk, the line radial profile underestimates Σgas, as 13CO emission becomes optically thick. The same happens at large radii where the column densities become too low and 13CO is not able to efficiently self-shield. Moreover, the disk becomes too cold and a considerable fraction of 13CO is frozen out, thus it does not contribute to the line emission. If the gas surface density profile is a simple power-law of the radius, the input power-law index can be retrieved within a ~20% uncertainty if one choses the proper radial range. If instead Σgas(R) follows the self-similar solution for a viscously evolving disk, retrieving the input power-law index becomes challenging, in particular for small disks. Nevertheless, we find that the power-law index γ can be in any case reliably fitted at a given line intensity contour around 6 K km s−1, and this produces a practical method to constrain the slope of Σgas(R). Application of such a method is shown in the case study of the TW Hya disk. Conclusions. Spatially resolved 13CO line radial profiles are promising to probe the disk surface density distribution, as they directly trace Σgas(R) profile at radii well resolvable by ALMA. There, chemical processes like freeze-out and isotope-selective photodissociation do not affect the emission, and, assuming that the volatile carbon does not change with radius, no chemical model is needed when interpreting the observations.

Secular Evolution Driven by Massive Eccentric Disks/Rings: An Apsidally Aligned Case

Abstract Massive eccentric disks (gaseous or particulate) orbiting a dominant central mass appear in many astrophysical systems, including planetary rings, protoplanetary and accretion disks in binaries, and nuclear stellar disks around supermassive black holes in galactic centers. We present an analytical framework for treating the nearly Keplerian secular dynamics of test particles driven by the gravity of an eccentric, apsidally aligned, zero-thickness disk with arbitrary surface density and eccentricity profiles. We derive a disturbing function describing the secular evolution of coplanar objects, which is explicitly related (via one-dimensional, convergent integrals) to the disk surface density and eccentricity profiles without using any ad hoc softening of the potential. Our analytical framework is verified via direct orbit integrations, which show it to be accurate in the low-eccentricity limit for a variety of disk models (for disk eccentricity ≲0.1–0.2). We find that free precession in the potential of a disk with a smooth surface density distribution can naturally change from prograde to retrograde within the disk. Sharp disk features—edges and gaps—are the locations where this tendency is naturally enhanced, while the precession becomes very fast. Radii where free precession changes sign are the locations where substantial (formally singular) growth of the forced eccentricity of the orbiting objects occurs. Based on our results, we formulate a self-consistent analytical framework for computing an eccentricity profile for an aligned, eccentric disk (with a prescribed surface density profile) capable of precessing as a solid body under its own self-gravity.

Multiple Gaps in the Disk of the Class I Protostar GY 91

Abstract We present the highest spatial resolution Atacama Large Millimeter/submillimeter Array (ALMA) observations to date of the Class I protostar GY 91 in the ρ Ophiuchus L1688 molecular cloud complex. Our 870 μm and 3 mm dust continuum maps show that the GY 91 disk has a radius of ∼80 au, and an inclination of ∼40°, but most interestingly that the disk has three dark lanes located at 10, 40, and 70 au. We model these features assuming they are gaps in the disk surface density profile and find that their widths are 7, 30, and 10 au. These gaps bear a striking resemblance to the gaps seen in the HL Tau disk, suggesting that there may be Saturn-mass planets hiding in the disk. To constrain the relative ages of GY 91 and HL Tau, we also model the disk and envelope of HL Tau and find that they are of similar ages, although GY 91 may be younger. Although snow lines and magnetic dead zones can also produce dark lanes, if planets are indeed carving these gaps then Saturn-mass planets must form within the first ∼0.5 Myr of the lifetime of protoplanetary disks.

Stability aspects of relativistic thin magnetized disks

We adapt the well known "displace, cut and reflect" method to construct exact solutions of the Einstein-Maxwell equations corresponding to infinitesimally thin disks of matter endowed with dipole magnetic fields, which are entirely supported by surface polar currents on the disk. Our starting point is the Gutsunaev-Manko axisymmetric solution describing massive magnetic dipoles in General Relativity, from which we obtain a continuous three-parameter family of asymptotically flat static magnetized disks with finite mass and energy. For strong magnetic fields, the disk surface density profile resembles some well known self-gravitating ring-like structures. We show that many of these solutions can be indeed stable and, hence, they could be in principle useful for the study of the abundant astrophysical situations involving disks of matter and magnetic fields.

THE IMPACT OF STELLAR FEEDBACK ON THE STRUCTURE, SIZE, AND MORPHOLOGY OF GALAXIES IN MILKY-WAY-SIZED DARK MATTER HALOS

ABSTRACT We use cosmological zoom-in simulations of galaxy formation in a Milky-Way-sized halo started from identical initial conditions to investigate the evolution of galaxy sizes, baryon fractions, morphologies, and angular momenta in runs with different parameters of the star formation–feedback cycle. Our fiducial model with a high local star formation efficiency, which results in efficient feedback, produces a realistic late-type galaxy that matches the evolution of basic properties of late-type galaxies: stellar mass, disk size, morphology dominated by a kinematically cold disk, stellar and gas surface density profiles, and specific angular momentum. We argue that feedback’s role in this success is twofold: (1) removal of low angular momentum gas, and (2) maintaining a low disk-to-halo mass fraction, which suppresses disk instabilities that lead to angular momentum redistribution and a central concentration of baryons. However, our model with a low local star formation efficiency, but large energy input per supernova, chosen to produce a galaxy with a similar star formation history as our fiducial model, leads to a highly irregular galaxy with no kinematically cold component, overly extended stellar distribution, and low angular momentum. This indicates that only when feedback is allowed to become vigorous via locally efficient star formation in dense cold gas do resulting galaxy sizes, gas/stellar surface density profiles, and stellar disk angular momenta agree with observed z = 0 galaxies.

FORMING DISK GALAXIES IN WET MAJOR MERGERS. I. THREE FIDUCIAL EXAMPLES

ABSTRACT Using three fiducial N-body+SPH simulations, we follow the merging of two disk galaxies that each have a hot gaseous halo component, and examine whether the merger remnant can be a spiral galaxy. The stellar progenitor disks are destroyed by violent relaxation during the merging and most of their stars form a classical bulge, while the remaining stars, as well as stars born during the merging times, form a thick disk and its bar. A new stellar disk forms subsequently and gradually in the remnant from the gas accreted mainly from the halo. It is vertically thin and well extended in its equatorial plane. A bar starts forming before the disk is fully in place, which is contrary to what is assumed in idealized simulations of isolated bar-forming galaxies, and has morphological features such as ansae and boxy/peanut bulges. Stars of different ages populate different parts of the box/peanut. A disky pseudobulge also forms, so that by the end of the simulation all three types of bulges coexist. The oldest stars are found in the classical bulge, followed by those of the thick disk, then by those in the thin disk. The youngest stars are in the spiral arms and the disky pseudobulge. The disk surface density profiles are of type II (exponential with downbending); the circular velocity curves are flat and show that the disks are submaximum in these examples: two clearly so and one near-borderline between maximum and submaximum. On average, only roughly between 10% and 20% of the stellar mass is in the classical bulge of the final models, i.e., much less than in previous simulations.

Formation of terrestrial planets in disks with different surface density profiles

We present the results of an extensive study of the final stage of terrestrial planet formation in disks with different surface density profiles and for different orbits of Jupiter and Saturn. We carried out simulations for disk densities proportional to r^-0.5, r^-1, and r^-1.5, and also for partially depleted disks as in the recent model of Mars formation by Izidoro et al (2014). The purpose of our study is to determine how the final assembly of planets and their physical properties are affected by the total mass of the disk and its radial profile. Because of the important roles of secular resonances in orbits and properties of the final planets, we studied the effects of these resonances as well. We have divided this study into two parts. In Part 1, we are interested in examining the effects of secular resonances on the formation of Mars and orbital stability of terrestrial planets. In Part 2, our goal is to determine trends that may exist between the disk surface density profile and the final properties of terrestrial planets. In the context of the depleted disk model, results show that the nu_5 resonance does not have a significant effect on the final orbits of terrestrial planets. However, nu_6 and nu_16 resonances play important roles in clearing their affected areas ensuring that no additional mass will be scattered into the accretion zone of Mars so that it can maintain its mass and orbital stability. In Part 2, our results indicate that despite some small correlations, in general, no trend seems to exist between the disk surface density profile and the mean number of the final planets, their masses, time of formation, and distances to the central star. We present the results of our simulations and discuss their implications for the formation of Mars and other terrestrial planets, as well as the physical properties of these objects such as their masses and water contents.

THE KOZAI–LIDOV MECHANISM IN HYDRODYNAMICAL DISKS. II. EFFECTS OF BINARY AND DISK PARAMETERS

Martin et al. (2014b) showed that a substantially misaligned accretion disk around one component of a binary system can undergo global damped Kozai-Lidov oscillations. During these oscillations, the inclination and eccentricity of the disk are periodically exchanged. However, the robustness of this mechanism and its dependence on the system parameters were unexplored. In this paper, we use three-dimensional hydrodynamical simulations to analyze how various binary and disk parameters affect the Kozai-Lidov mechanism in hydrodynamical disks. The simulations include the effect of gas pressure and viscosity, but ignore the effects of disk self-gravity. We describe results for different numerical resolutions, binary mass ratios and orbital eccentricities, initial disk sizes, initial disk surface density profiles, disk sound speeds, and disk viscosities. We show that the Kozai-Lidov mechanism can operate for a wide range of binary-disk parameters. We discuss the applications of our results to astrophysical disks in various accreting systems.

Triggering active galactic nuclei in hierarchical galaxy formation: disk instability vs. interactions

Using a semi analytic model for galaxy formation we investigate the effects of Black Hole accretion triggered by disk instabilities (DI) in isolated galaxies on the evolution of AGN. Specifically, we took on, developed and expanded the Hopkins & Quataert (2011) model for the mass inflow following disk perturbations, and compare the corresponding evolution of the AGN population with that arising in a scenario where galaxy interactions trigger AGN (IT mode). We extended and developed the DI model by including different disk surface density profiles, to study the maximal contribution of DI to the evolution of the AGN population. We obtained the following results: i) for luminosities corresponding to $M_{1450}\gtrsim -26$ the DI mode can provide the BH accretion needed to match the observed AGN luminosity functions up to $z \approx 4.5$; in such a luminosity range and redshift, it can compete with the IT scenario as the main driver of cosmological evolution of AGN; ii) The DI scenario cannot provide the observed abundance of high-luminosity QSO with $M_{1450}\lesssim -26$ AGN, as well as the abundance of high-redhshift $z \approx 4.5$ QSOs with $M_{1450}\lesssim -24$, while the IT scenario provides an acceptable match up to $z \approx 6$, as found in our earliest works; iii) The dispersion of the distributions of Eddington ratio for low- and intermediate-luminosity AGN (bolometric $L_{AGN}$ = $10^{43}$ - $10^{45}$ erg/s) is predicted to be much smaller in the DI scenario compared to the IT mode; iv) The above conclusions are robust with respect to the explored variants of the Hopkins & Quataert (2011) model. We discuss the physical origin of our findings, and how it is possible to pin down the dominant fueling mechanism in the low-intermediate luminosity range $M_{1450}\gtrsim -26$ where both the DI and the IT modes are viable candidates as drivers for the AGN evolution.

DEAD, UNDEAD, AND ZOMBIE ZONES IN PROTOSTELLAR DISKS AS A FUNCTION OF STELLAR MASS

We investigate the viability of the magnetorotational instability (MRI) in accretion disks around both solar-type stars and very low mass stars. In particular, we determine the disk regions where the MRI can be shut off either by Ohmic resistivity (the so-called Dead and Undead Zones) or by ampipolar diffusion (a region we term the Zombie Zone). We consider 2 stellar masses: Mstar = 0.7 and 0.1 Msun. In each case, we assume that: the disk surface density profile is that of a scaled Minimum Mass Solar Nebula, with Mdisk/Mstar ~ 0.01 as currently estimated; disk ionisation is driven primarily by stellar X-rays, complemented by cosmic rays and radionuclides; and the stellar X-ray luminosity scales with bolometric luminosity as Lx/Lstar ~ 10^-3.5, as observed. Ionization rates are calculated with the MOCCASIN code, and ionisation balance determined using a simplified chemical network, including well-mixed 0.1 um grains at various levels of depletion. We find that (1) ambipolar diffusion is the primary factor controlling MRI activity in disks around both solar-type and very low mass stars. Assuming that the MRI yields the maximum possible field strength at each radius, we further find that: (2) the MRI-active layer constitutes only ~ 5-10% of the total disk mass; (3) the accretion rate (Mdot) varies radially in both magnitude and sign (inward or outward), implying time-variable accretion as well as the creation of disk gaps and overdensities, with consequences for planet formation and migration; (4) achieving the empirical accretion rates in solar-type and very low mass stars requires a depletion of well-mixed small grains by a factor of 10-1000 relative to the standard dust-to-gas mass ratio of 10^-2; and (5) the current non-detection of polarized emission from field-aligned grains in the outer disk regions is consistent with active MRI at those radii.

Planetesimal formation by turbulent concentration

The formation of 1–1000 km diameter planetesimals from dust grains in a protoplanetary disk is a key step in planet formation. Conventional models for planetesimal formation involve pairwise sticking of dust grains, or the sedimentation of dust grains to a thin layer at the disk midplane followed by gravitational instability. Each of these mechanisms is likely to be frustrated if the disk is turbulent. Particles with stopping times comparable to the turnover time of the smallest eddies in a turbulent disk can become concentrated into dense clumps that may be the precursors of planetesimals. Such particles are roughly millimeter-sized for a typical protoplanetary disk. To survive to become planetesimals, clumps need to form in regions of low vorticity to avoid rotational breakup. In addition, clumps must have sufficient self gravity to avoid break up due to the ram pressure of the surrounding gas. Given these constraints, the rate of planetesimal formation can be estimated using a cascade model for the distribution of particle concentration and vorticity within eddies of various sizes in a turbulent disk. We estimate planetesimal formation rates and planetesimal diameters as a function of distance from a star for a range of protoplanetary disk parameters. For material with a solar composition, the dust-to-gas ratio is too low to allow efficient planetesimal formation, and most solid material will remain in small particles. Enhancement of the dust-to-gas ratio by 1–2 orders of magnitude, either vertically or radially, allows most solid material to be converted into planetesimals within the typical lifetime of a disk. Such dust-to-gas ratios may occur near the disk midplane as a result of vertical settling of short-lived clumps prior to clump breakup. Planetesimal formation rates are sensitive to the assumed size and rotational speed of the largest eddies in the disk, and formation rates increase substantially if the largest eddies rotate more slowly than the disk itself. Planetesimal formation becomes more efficient with increasing distance from the star unless the disk surface density profile has a slope of −1.5 or steeper as a function of distance. Planetesimal formation rates typically increase by an order-of-magnitude or more moving outward across the snow line for a solid surface density increase of a factor of 2. In all cases considered, the modal planetesimal size increases with roughly the square root of distance from the star. Typical modal diameters are 100 km and 400 km in the regions corresponding to the asteroid belt and Kuiper belt in the Solar System, respectively.

THE SIGNATURE OF THE ICE LINE AND MODEST TYPE I MIGRATION IN THE OBSERVED EXOPLANET MASS-SEMIMAJOR AXIS DISTRIBUTION

Existing exoplanet radial velocity surveys are complete in the planetary mass-semimajor axis (Mp-a) plane over the range 0.1 AU < a < 2.0 AU where Mp >~ 100 M_Earth. We marginalize over mass in this complete domain of parameter space and demonstrate that the observed semimajor axis distribution is inconsistent with models of planet formation that use the full Type I migration rate derived from a linear theory and that do not include the effect of the ice line on the disk surface density profile. However, the efficiency of Type I migration can be suppressed by both nonlinear feedback and the barriers introduced by local maxima in the disk pressure distribution, and we confirm that the synthesized Mp-a distribution is compatible with the observed data if we account for both retention of protoplanetary embryos near the ice line and an order-of-magnitude reduction in the efficiency of Type I migration. The validity of these assumption can be checked because they also predict a population of short-period rocky planets with a range of masses comparable to that of the Earth as well as a "desert" in the Mp-a distribution centered around Mp ~ 30-50 M_Earth and a < 1 AU. We show that the expected "desert" in the Mp-a plane will be discernible by a radial velocity survey with 1 m/s precision and n ~ 700 radial velocity observations of program stars.

Baryonic Collapse within Dark Matter Halos and the Formation of Gaseous Galactic Disks

This paper constructs an analytic framework for calculating the assembly of galactic disks from the collapse of gas within dark matter halos, with the goal of determining the resulting surface density profiles. In this formulation, gas parcels (baryons) fall through the potentials of dark matter halos on nearly ballistic, zero-energy orbits and collect in a rotating disk. The dark matter halos have a nearly universal form, as determined previously through numerical simulations. The simplest scenario starts with a gaseous sphere in slow uniform rotation, follows its subsequent collapse, and determines the surface density of the disk. This calculation is carried out for precollapse mass distributions of the form M(r) ~ rp and for polytropic spheres in hydrostatic equilibrium with the halo potential. The resulting disk surface density profiles have nearly power-law forms σ ~ -q, with well-defined edges determined by the centrifugal barrier RC, the radius to which gas with the highest specific angular momentum falls during the collapse. This scenario is generalized to include nonspherical starting states, alternate rotation profiles, and multiple accretion events (e.g., due to gas added via merger events). This latter complication considers a lognormal distribution for the angular momenta of the precollapse states of the individual components. In the limit where this distribution is wide, the composite surface density approaches a universal form σT ~ -2, independent of the shape of the constituent profiles. When the angular momentum distribution has an intermediate width, however, the composite surface density attains a nearly exponential form, roughly consistent with profiles of observed galaxies.